Applied Economics, 1996, 28, 1191Ð 1198
Portfolio analysis with a large universe
of assets
DAVID NAWROCKI
College of Commerce and Finance, 800 Lancaster Avenue, Villanova University,
Villanova, PA 19085, USA
Covariance matrix optimization algorithms are applied to a large number of assets.
A previous paper by Burgess and Bey (1988) suggests that attempting to optimize
a large number of securities with the traditional covariance matrix model is not
practical. An alternative approach ranks the securities with the reward to volatility
(reward to beta) ratio and then optimizes a smaller subset of securities with the
covariance matrix model. This study proposes additional screening methods such as
stochastic dominance, reward to variability (R/V) ratios, reward to lower partial
moment (R/LPM) ratios, and the optimization of subgroups, and provides an empirical test of the various screening methodologies. The results indicate that the full
covariance critical line optimization algorithm is surprisingly robust compared to the
other techniques.
I. INTRODUCTION
Covariance matrix optimization algorithms are applied to
a large number of assets. The purpose is to test the robustness of the covariance optimization algorithm under these
conditions. Because of the computational complexity of the
covariance model, Burgess and Bey (1988) suggest using
a combination of the Elton, Gruber and Padberg (1976)
single-index model (SIM) (to screen assets) and the Markowitz (1959) covariance model (to optimize portfolios) in
order to handle a large number of assets. This study proposes additional screening methods such as reward to
variability (R/V) ratios, reward to lower partial moment
(R/LPM) ratios, stochastic dominance and subgroup optimization, and provides an empirical test of the various
screening methods. Surprisingly, the critical line covariance
algorithm is very robust with ex ante data and there seems
little need to replace it. This result is in line with Markowitz
(1987), who states that the critical line algorithm is successful at handling various problems caused by a large universe
of securities.
Section II summarizes the problem of analysing large
numbers of securities with portfolio optimization techniques and suggests alternative methodologies. Section III
discusses the methodology of the empirical test and Section
0003Ð 6846
Ó
1996 Routledge
IV presents the empirical results. Section V summarizes and
concludes the paper.
I I . R E V I E W O F T H E LI T E R A T U R E
Applying portfolio-theoretic models to a large number of
assets is problematic. The computational complexity of the
Markowitz covariance matrix approach grows exponentially as the number of assets increases, thus increasing
computational time and resources. With 100 assets, it is
reasonable for a portfolio management team to develop
expected returns and expected variances for 100 securities.
However, it is unrealistic to expect the team to consider
carefully, one at a time, the 4950 covariances needed to
complete the model. The use of historical covariances as
estimates of expected covariances has at least two problems.
First, unaltered history may lack information currently
known about the status of speci® c securities, and second, the
number of periods of relevant data may be less than the
number of securities analysed.
The problem is that the ® nancial literature usually assumes the covariance matrix to be non-singular or positive
de® nite. However, the covariance matrix may be positive
semi-de® nite, allowing for both singular and non-singular
1191
1192
D. Nawrocki
matrices. It is important to handle singular matrices because
there are applications where the covariance matrix is singular and a resulting solution would be useful. Markowitz
(1987) describes three such cases of singular matrices.
First, the estimation of historical covariances may use
fewer observations than there are securities. This is the
invertibility of the matrix problem that results in a singular
matrix. Markowitz (1987) notes that the critical line algorithm for computing mean-variance (EV) e cient sets still
succeeds and most of the geometric facts about e cient sets
hold true for singular matrices.
Second, the addition of constraints to the portfolio problem adds slack variables that will provide a singular matrix.
Third, buying a call option and selling a put option are
perfectly correlated with the underlying security. In addition, long and short margin positions will also provide
perfect positive and negative correlations and, therefore,
a singular matrix.
The focus of this paper is to explore the ® rst case of
optimizing a large number of assets with limited historic
observations. Increasing the sample size to solve the ® rst case
is not the obvious answer that it seems. A number of studies
conclude that estimates of asset variability are non-stationary
(MacKinlay, 1987; Schwert, 1989; and Schwert and Seguin,
1990). These studies indicate that historic periods should not
exceed six to seven years of data (84 periods of monthly data),
thus limiting the number of assets in the analysis.
Another solution is to substitute a diagonal matrix model
as suggested by Markowitz (1959) and Sharpe (1963) that
avoids the invertibility of the matrix problem. Cohen and
Pogue (1967) demonstrate that the resulting single-index
model (SIM) performs as well as the covariance model.
However, King (1966) and Rosenberg (1974) found con¯ icting results. The SIM model provides an optimal answer as
long as error terms in the Sharpe regressions are crosssectionally independent (residual covariances are zero).
More recently, Hammer and Phillips (1992) demonstrated
(when applying the SIM) that signi® cant residual covariances exist, which results in super¯ uous (and ine cient) diversi® cation.
Table 1. Comparison of Markowitz critical line algorithm and Martin simultaneous equation algorithm for di¤ erent size security universes
(monthly returns in %). Highest R/V portfolio from Markowitz e¦ cient frontier is reported for each security universe
Markowitz critical line
N
Mean
Std dev.
R/V
Martin simultaneous equations
M
Mean
Std dev.
R/V
M
% di€ erence
5
10
7
9
12
13
14
16
17
17
17
18
1.976
1.836
2.197
2.123
2.032
2.103
1.997
2.060
2.077
2.071
1.982
2.033
3.593
3.593
3.239
3.019
2.877
3.132
3.165
3.381
3.589
2.909
3.043
2.950
0.5499
0.5709
0.6782
0.7033
0.7061
0.6712
0.6308
0.6093
0.5787
0.7122
0.6515
0.6890
5
10
7
9
10
9
9
5
5
11
11
9
0.00
0.00
0.00
0.00
15.30
40.79
67.96
66.22
78.26
35.30
57.49
61.05
6
8
10
10
11
11
13
14
9
9
9
11
1.413
1.586
1.665
1.802
1.789
1.812
1.825
1.790
1.853
1.853
1.849
1.828
3.558
3.713
3.730
3.643
3.597
3.574
3.648
3.585
3.458
3.415
3.442
3.369
0.3972
0.4270
0.4465
0.4946
0.4974
0.5071
0.5002
0.4993
0.5356
0.5426
0.5372
0.5426
6
8
8
9
11
10
12
10
8
9
11
8
0.00
0.00
11.36
13.39
12.49
2.83
18.58
25.16
16.56
2.73
13.51
17.59
66 monthly returns, January 1980 Ð June 1985
10
20
30
40
50
60
70
80
90
100
110
120
1.976
1.836
2.197
2.123
2.032
2.103
1.997
2.060
2.077
2.071
1.982
2.033
3.593
3.216
3.239
3.019
2.838
2.918
2.746
2.810
2.810
2.809
2.602
2.662
0.5499
0.5709
0.6782
0.7033
0.7159
0.7205
0.7273
0.7330
0.7392
0.7373
0.7620
0.7635
120 monthly returns, August 1981 Ð July 1991
10
20
30
40
50
60
70
80
90
100
110
120
1.413
1.586
1.665
1.802
1.789
1.812
1.825
1.790
1.853
1.853
1.849
1.828
3.558
3.713
3.703
3.627
3.581
3.584
3.607
3.538
3.410
3.410
3.406
3.336
0.3972
0.4270
0.4496
0.4969
0.4996
0.5057
0.5059
0.5059
0.5434
0.5434
0.5429
0.5478
N Ð Number of stocks in universe
M Ð Number of stocks in Markowitz/Martin e cient portfolio
% di€ erence Ð proportion of the portfolio that has to be bought/sold to move from Markowitz allocations to Martin allocations.
1193
Portfolio analysis with a large universe of assets
The problem with the size of the covariance matrix is
evident in Table 1. Two alternative covariance model optimization algorithms generate comparable portfolios from
the same set of covariance matrices, varying only the number of assets in the matrix. The Markowitz (1959) critical
line algorithm generates corner portfolios along the e cient
frontier. The corner portfolio with the highest reward to
variability (R/V) ratio compares to a portfolio generated by
the calculus-minimization simultaneous equations algorithm developed by Martin (1955). This algorithm minimizes the portfolio variance for a speci® ed return, which in
this case is the expected return of the highest R/V ratio
portfolio from the Markowitz critical line algorithm. The
Martin simultaneous equations approach is in numerous
books including Francis and Archer (1979) and Elton and
Gruber (1987). The calculus-minimization approach uses
Lagrangain multipliers in order to handle the constraints to
the problem. Thus, the introduction of slack variables into
the problem can create singular matrices.
The test runs two samples of 66 and 120 months. The
number of securities in the analysis increases from ten to 120
securities.
With 66 monthly returns, the two algorithms return
equivalent portfolios until the size of the covariance matrix
reaches 50 assets. At this point, the Martin algorithm is less
e cient (by providing higher variances and lower reward to
variability (R/V) ratios). As the number of assets increases
past 50, 35 to 78% of the portfolio allocations generated by
the two algorithms are di€ erent. In addition, the Martin
algorithm is providing signi® cantly smaller portfolios in
terms of the number of securities. Clearly the Martin algorithm is sensitive to the singular covariance matrix problem,
possibly because of the Lagrangian multipliers.
With 120 monthly returns, di€ erences between the two
algorithms appear with 30 assets in the covariance matrix.
However, the di€ erences only a€ ect 3 to 25% of the portfolio allocations as the number of assets increases to 120.
Also, there is not the drastic di€ erence in the number of
securities in a portfolio or the R/V ratios observed in the
66-month results. Increasing the number of observations
from 66 to 120 months provides better correspondence
between the two algorithms.
The critical line algorithm seems to be very robust with
the singular-matrix problem. However, because of the divergence of results, as the number of assets increases past 30,
the question arises if the critical line algorithm is providing
optimal results with a large number of assets in the
covariance matrix. Burgess and Bey (1988) discuss one possible solution that uses the Elton, Gruber and Padberg
(1976) SIM (hereafter, EGP) algorithm. This algorithm
ranks assets using reward to volatility ratios (measured by
beta, or an R/B ratio) and then computes optimal weights.
Ranking with the R/B ratio handles a large number of
assets. Then the securities are tested for entry into the EGP
portfolio. The assets that qualify become the input for the
Markowitz (1959) critical line algorithm. Since a smaller
number of assets are in the resulting covariance matrix, this
process reduces the invertibility of matrix problem and
eliminates the residual covariance problem.
Along the same lines, Nawrocki (1990) uses the reward to
lower partial moment (R/LPM) ratio as a simple heuristic
for selecting portfolios. These contributions open other
ranking possibilities that use stochastic dominance, reward
to variability ratios and subgroup optimization to screen
assets before a ® nal optimization with the covariance matrix
analysis.
III. METHODOLOGY
This paper provides a test of various screening strategies
that attempt to minimize the singular-matrix problem by
reducing the number of assets in the covariance matrix. The
main test is a comparison of ex ante (or historic) performance of these strategies with the optimization of all the
assets in one large covariance matrix.
The data samples used in the study include 125 common
stocks randomly selected from the 1992 CRSP data tape.
The two data ® les contain di€ erent stocks. One contains 120
monthly total returns from August 1981 to July 1991. The
second contains 66 monthly total returns from January
1980 to June 1985. (The latter time period corresponds to
the same data period used in the Burgess and Bey, 1988
study.) Given the previously cited research, the following
techniques suggest themselves as potential solutions to the
singular-matrix problem. The full covariance algorithms
(Markowitz and Martin) provide the benchmark performance since the purpose of the paper is to study the robustness of the Markowitz critical line algorithm with a large
number of assets.
(1) Markowitz (critical line) and Martin (simultaneous
equations) optimization of all 125 assets.
(2) Markowitz critical line optimization of subgroups of
25 assets. Securities that do not enter the subgroup
portfolio do not appear in future optimizations. The
process continues until approximately 25 assets remain and the ® nal optimization takes place.
(3) Markowitz critical line optimization of subgroups of
50 assets at a time.
(4) Second degree (SSD) and third degree (TSD) stochastic dominance techniques are used to screen the 125
assets to the SSD and TSD e cient sets. The resulting
group of assets become the input to the Markowitz
critical line optimization. The stochastic dominance
uses the Porter, Wart and Ferguson (1973) algorithm
as corrected by Vickson and Altman (1977).
(5) Reward to LPM (R/LPM) ratios screen the 125 assets.
The top 25 assets become the input to the Markowitz
critical line optimization. The screening algorithm
1194
D. Nawrocki
Table 2. Summary statistics comparison of di¤ erent portfolio selection algorithms for 66 monthly observations (January 1980 Ð June 1985)
Method
Mean
Std dev
Semi-dev
R/SV
R/V
SSD
TSD
Markowitz (125)
Martin (125)
EV Group 25
EV Group 50
SSD-EV
TSD-EV
R/LPM n = 1 EV
R/LPM n = 2 EV
R/LPM n = 3 EV
R/V EV
R/B EV
R/B EGP SI
R/V EGP AC
R/LPM n = 1 R/LPM
R/LPM n = 2 R/LPM
R/LPM n = 3 R/LPM
R/LPM n = 1 LPM QP
R/LPM n = 2 LPM QP
R/LPM n = 3 LPM QP
2.0182
2.0324
2.0489
2.0921
2.3870
2.4129
2.1330
2.0462
2.2043
2.1398
2.2754
2.3032
2.2411
2.1542
2.1425
2.1455
2.1254
2.1761
2.1129
2.6613
2.9856
2.7129
2.7922
3.6085
3.7312
3.1214
3.0084
3.2613
3.1635
3.7571
3.8660
3.6009
3.4871
3.4875
3.4800
3.2239
3.2940
3.1013
0.7930
0.9756
0.7998
0.8628
1.1667
1.2153
0.9999
0.8939
0.9946
0.9842
1.1843
1.1976
1.1298
1.2648
1.2053
1.1701
0.9924
0.9519
0.9434
2.5449
2.0831
2.5616*
2.4245
2.0459
1.9853
2.1331
2.2889
2.2161
2.1741
1.9213
1.9231
1.9835
1.7031
1.7775
1.8335
2.1416
2.2858
2.2395
0.7583
0.6807
0.7552
0.7492
0.6615
0.6467
0.6833
0.6801
0.6759
0.6764
0.6056
0.5957
0.6223
0.6177
0.6143
0.6165
0.6592
0.6606
0.6813
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Std dev Ð Standard deviation of portfolio
Semi-dev Ð Semi-deviation of portfolio (square root of semi-variance)
R/SV Ð Reward to semi-variability ratio (LPM degree n = 2.0)
R/B Ð Reward to beta ratio
R/V Ð Reward to variability ratio
R/LPM n = 1 Ð Reward to lower partial moment, degree n = 1.0
SSD Ð Second degree dominance e cient sets
TSD Ð Third degree dominance e cient sets
EV Ð Markowitz expected return-variance critical line algorithm
EGP SI Ð Elton, Gruber, Padberg single-index algorithm
EGP AC Ð Elton, Gruber, Padberg average correlation algorithm
LPM QP Ð Co-LPM quadratic programming (critical line) algorithm
R/LPM Ð Nawrocki reward/LPM heuristic algorithm
* Indicates performance superior to Markowitz 125-asset optimization.
utilizes LPM degrees of 1,2 and 3. The degrees of the
LPM are a measure of investor risk-aversion. An
LPM degree of 1 indicates a risk-neutral investor.
Increasing the LPM degree from 1 to 2 or 3 indicates
increasing risk-aversion by the investor (See Fishburn, 1977 for more information on this matter).
(6) Reward to variability (R/V) ratios screen the 125
assets. The top 25 assets become the input to the
Markowitz critical line optimization and the Elton,
Gruber and Padberg (1976) average correlation heuristic algorithm.
(7) Reward to volatility (R/B) ratios screen the 125 assets.
The Elton, Gruber and Padberg (1976) SIM and the
Markowitz critical line covariance matrix models (the
latter is the Burgess and Bey, 1988 approach) optimize the assets that qualify.
1
(8) R/LPM ratios screen assets. Computation of portfolios uses an R/LPM heuristic (Nawrocki, 1983,
1990; Nawrocki and Staples, 1989) and a co-LPM
matrix is optimized using the Markowitz critical line
algorithm (Nawrocki, 1991, 1992).
Both the reward to semivariability (R/SV) and reward to
variability (R/V) ratios evaluate the resulting portfolios.
However, numerous studies have found statistical biases in
the traditional Sharpe (R/V), Treynor (R/B), and Jensen
measures. 1 Ang and Chua (1979) demonstrate that the R/SV
ratio provides signi® cantly less biased ranking of portfolio
performance when compared to the traditional measures.
Therefore, the discussion of the empirical results places
more emphasis on the R/SV results.
The Sharpe performance measure (R/V ratio) dates back to Roy (1952). Sharpe stated in a speech to the Financial Management
Association in October 1993 that he refers to it as the reward to variability ratio. He does not feel that it is appropriate to call it the Sharpe
measure.
1195
Portfolio analysis with a large universe of assets
Table 3. Summary statistics comparison of di¤ erent portfolio selection algorithms for 120 monthly observations (August 1981 Ð July 1991)
Method
Mean
Std dev
Semi-dev
R/SV
R/V
SSD
Markowitz (125)
Martin (125)
EV Group 25
EV Group 50
SSD-EV
TSD-EV
R/LPM n = 1 EV
R/LPM n = 2 EV
R/LPM n = 3 EV
R/V EV
R/B EV
R/B EGP SI
R/V EGP AC
R/LPM n = 1 R/LPM
R/LPM n = 2 R/LPM
R/LPM n = 3 R/LPM
R/LPM n = 1 LPM QP
R/LPM n = 2 LPM QP
R/LPM n = 3 LPM QP
1.9573
1.9700
1.9696
1.9674
1.9947
1.9870
1.9680
1.9671
1.9245
1.9690
1.9464
2.1447
1.8725
1.7651
1.7959
1.7772
1.9933
1.8483
1.8553
3.6811
4.1900
3.6786
3.6787
3.8552
3.8335
3.6956
3.6956
3.6957
3.6957
3.6571
4.5340
3.5749
3.6432
3.7012
3.7227
3.7895
3.4441
3.5627
1.8873
2.0319
1.8530
1.8538
1.8646
1.8584
1.8454
1.8457
1.7933
1.8451
1.8266
2.1508
1.7503
1.8626
1.8187
1.8105
1.8657
1.7189
1.7143
1.0370
0.9695
1.0629 *
1.0612 *
1.0697 *
1.0691 *
1.0664 *
1.0657 *
1.0731 *
1.0671 *
1.0655 *
0.9971
1.0698 *
0.9476
0.9874
0.9816
1.0683 *
1.0752 *
1.0828 *
0.5317
0.4702
0.5354 *
0.5348 *
0.5174
0.5183
0.5325 *
0.5322 *
0.5288
0.5328 *
0.5322 *
0.4730
0.5238
0.4845
0.4852
0.4774
0.5260
0.5366 *
0.5210
X
TSD
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Std dev Ð Standard deviation of portfolio
Semi-dev Ð Semi-deviation of portfolio (square root of semivariance)
R/SV Ð Reward to semi-variability ratio (LPM degree n = 2.0)
R/B Ð Reward to beta ratio
R/V Ð Reward to variability ratio
R/LPM n = 1 Ð Reward to lower partial moment, degree n = 1.0
SSD Ð Second degree dominance e cient portfolios
TSD Ð Third degree dominance e cient portfolios
EV Ð Markowitz expected return-variance critical line algorithm
EGP SI Ð Elton, Gruber, Padberg single-index algorithm
EGP AC Ð Elton, Gruber, Padberg average correlation algorithm
LPM QP Ð Co-LPM quadratic programming (critical line) algorithm
R/LPM Ð Nawrocki reward/LPM heuristic algorithm
* Indicates performance superior to Markowitz 125-asset optimization.
IV. EMPIRICAL RESULT S
Tables 2Ð 5 provide the summary results for this study.
Tables 2 and 3 provide the results for the 66-month and
120-month samples, respectively. As in Table 1, the report
utilizes the maximum R/V portfolio for each technique.
With both of these sample periods, the Markowitz 125-asset
optimization is very impressive. With the 66-month sample,
only the optimization of subgroups of 25 and 50 assets (EV
Group 25 and EV Group 50) is competitive with the 125asset Markowitz analysis. The R/LPM screening with
mean-variance (EV) optimization and with mean-lower
partial moment (LPM QP) optimization and the R/V
screening with mean-variance (EV) optimization are the
only other alternatives that are close to the full 125-asset
Markowitz optimization. The Martin 125-asset optimization is about the fourth best of the alternatives, thus con® rming the result from Table 1 that the simultaneous
equations approach does not handle large numbers of
assets as well as the Markowitz critical line algorithm.
Only the 25-asset subgrouping (EV Group 25) approach
outperforms the Markowitz 125 assets when using the
R/SV ratio to compare performance. The Markowitz critical line algorithm is also an undominated member of the
second (SSD) and third (TSD) degree stochastic dominance
sets.
Using 120 months, the alternative screening methods that
use mean-variance (EV) optimization or LPM optimization
(LPM QP), after ranking and grouping, exhibit higher R/SV
and R/V ratios than the Markowitz 125-asset optimization,
which comes next to last. The Elton, Gruber and Padberg
algorithms are not competitive. The Markowitz critical line
algorithm is a member of the SSD undominated set but is
not a member of the TSD undominated set. The results
indicate that the Markowitz critical line algorithm is more
robust with a small number of observations than the suggested alternatives. When the number of observations is
close to the number of assets, the alternatives are competitive with the Markowitz critical line algorithm and actually
dominate it in the TSD e cient set.
1196
D. Nawrocki
Table 4. Reward to semivariability ratios (R/SV ) for each portfolio selection algorithm for each data sub-period (monthly data)
Method
Months
1Ð 60
Months
61Ð 120
Months
1Ð 120
Months
1Ð 66
Average
SSD
Markowitz (125)
Martin (125)
EV Group 25
EV Group 50
SSD-EV
TSD-EV
R/LPM n = 1 EV
R/LPM n = 2 EV
R/LPM n = 3 EV
R/V EV
R/B EV
R/B EGP SI
R/V EGP AC
R/LPM n = 1 R/LPM
R/LPM n = 2 R/LPM
R/LPM n = 3 R/LPM
R/LPM n = 1 LPM QP
R/LPM n = 2 LPM QP
R/LPM n = 3 LPM QP
4.6073
2.4750
4.4899
4.4831
4.4354
2.6840
2.7592
3.0816
3.0792
3.0825
2.8709
2.5761
2.1994
2.3434
2.5726
2.6131
2.6186
3.0675
3.3065
0.9032
0.7816
0.9226*
0.9226*
0.9296*
0.9276*
0.9076*
0.9070*
0.9072*
0.9064*
0.9301*
0.8737
0.6869
0.6763
0.6678
0.6713
0.9230*
0.9405*
0.9491*
1.0370
0.9695
1.0629*
1.0612*
1.0697*
1.0691*
1.0664*
1.0657*
1.0731*
1.0671*
1.0655*
0.9971
1.0698*
0.9476
0.9874
0.9816
1.0683*
1.0752*
1.0828*
2.5449
2.0831
2.5616*
2.4245
2.0459
1.9853
2.1331
2.2889
2.2161
2.1741
1.9213
1.9213
1.9835
1.7031
1.7775
1.8335
2.1416
2.2858
2.2395
2.2731
1.5773
2.2593
2.2229
2.1202
1.6665
1.7166
1.8358
1.8189
1.8075
1.6970
1.5905
1.4849
1.4176
1.5013
1.5249
1.6879
1.8423
1.8945
XXXX
X
XXXX
XXXX
XXXX
XXXX
XXXX
XXXX
XXXX
XX
XXXX
XXXX
XXXX
XX
XX
X
XXXX
XXXX
XXXX
R/SV Ð Reward to semivariability ratio (LPM degree n = 2.0)
R/B Ð Reward to beta ratio
R/V Ð Reward to variability ratio
R/LPM n = 1 Ð Reward to lower partial moment, degree n = 1.0
SSD Ð Second degree dominance e cient portfolios
TSD Ð Third degree dominance e cient portfolios
EV Ð Markowitz expected return-variance critical line algorithm
EGP SI Ð Elton, Gruber, Padberg single-index algorithm
EGP AC Ð Elton, Gruber, Padberg average correlation algorithm
LPM QP Ð Co-LPM quadratic programming (critical line) algorithm
R/LPM Ð Nawrocki reward/LPM heuristic algorithm
* Indicates superior performance.
Tables 4 and 5 present the results to breaking the 120month period into two 60-month subperiods. Table 4 presents the R/SV results; Table 5 provides the R/V results.
Both the R/SV results and the R/V results indicate that the
optimization of subgroups of 25Ð 50 assets provides the most
consistent ex ante results. The R/SV ratios indicate that all
the ranking techniques using critical line mean-variance or
mean-LPM optimization are reasonable alternatives. The
Martin simultaneous equations approach provides the
worst portfolio results. Again the Markowitz critical line
algorithm is a member of the SSD undominated set for all
four data sets.
Note that the results obtained here are relevant only to
the historic (ex ante) performance of the portfolios. The
results indicate that the critical line algorithm is robust
enough to handle a large number of assets when a limited
number of observations are available. Purchasing the portfolios as an investment during a holding period (ex post)
may result in di€ erent performance. Both Elton, Gruber and
Urich (1978) and Nawrocki (1991) provide empirical
evidence that their simple heuristic algorithms will outperform the more complex covariance matrix optimization
algorithm during holding (ex post) periods.
V . S U MM A R Y A N D C O N C L U S I O N S
The surprising result from this study is how robust the
Markowitz (1959) optimization algorithm is for 125 assets.
The results of this algorithm are very competitive with
methodology designed to minimize the invertibility of the
matrix problem. On the other hand, the Martin (1955)
simultaneous equations covariance algorithm is very sensitive to the number of securities in the covariance matrix.
Because this algorithm is popular in portfolio theory texts
for its pedagogical accessibility, it is important not to use it
whenever the number of assets in the covariance matrix
exceeds 30. Breaking the sample of 125 assets into subgroups of 25 assets, then optimizing the subgroups, seems
to be the only alternative methodology that consistently
1197
Portfolio analysis with a large universe of assets
Table 5. Reward to variability ratios (R/V ) for each portfolio selection algorithm for each data sub-period (monthly data)
Method
Months
1Ð 60
Months
61Ð 120
Months
1Ð 120
Months
1Ð 66
Average
Markowitz (125)
Martin (125)
EV Group 25
EV Group 50
SSD-EV
TSD-EV
R/LPM n = 1 EV
R/LPM n = 2 EV
R/LPM n = 3 EV
R/V EV
R/B EV
R/B EGP SI
R/V EGP AC
R/LPM n = 1 R/LPM
R/LPM n = 2 R/LPM
R/LPM n = 3 R/LPM
R/LPM n = 1 LPM QP
R/LPM n = 2 LPM QP
R/LPM n = 3 LPM QP
1.0287
0.8147
1.0295*
1.0289*
1.0282
0.8431
0.9205
0.9450
0.9446
0.9295
0.9357
0.8152
0.8310
0.8129
0.8183
0.8418
0.8926
0.9175
0.9111
0.4656
0.3885
0.4686*
0.4689*
0.4669*
0.4674*
0.4625
0.4624
0.4624
0.4622
0.4674*
0.4466
0.3921
0.4018
0.3959
0.3966
0.4464
0.4559
0.4484
0.5317
0.4702
0.5354*
0.5348*
0.5174
0.5183
0.5325*
0.5322*
0.5288
0.5328*
0.5322*
0.4730
0.5238
0.4845
0.4852
0.4774
0.5260
0.5366*
0.5210
0.7583
0.6807
0.7552
0.7492
0.6615
0.6467
0.6833
0.6801
0.6759
0.6764
0.6056
0.5957
0.6223
0.6177
0.6143
0.6165
0.6592
0.6606
0.6813
0.6961
0.5885
0.6972*
0.6955
0.6685
0.6189
0.6497
0.6549
0.6529
0.6502
0.6352
0.5826
0.5923
0.5792
0.5784
0.5831
0.6311
0.6427
0.6405
R/B Ð Reward to beta ratio
R/V Ð Reward to variability ratio
R/LPM n = 1 Ð Reward to lower partial moment, degree n = 1.0
SSD Ð Second degree dominance e cient portfolios
TSD Ð Third degree dominance e cient portfolios
EV Ð Markowitz expected return-variance critical line algorithm
EGP SI Ð Elton, Gruber, Padberg single-index algorithm
EGP AC Ð Elton, Gruber, Padberg average correlation algorithm
LPM QP Ð Co-LPM quadratic programming (critical line) algorithm
R/LPM Ð Nawrocki reward/LPM heuristic algorithm
* Indicates superior performance.
improves on the Markowitz critical line optimization of 125
assets, and then only with a large number of observations.
The Burgess and Bey (1988) method of screening and then
optimizing is an acceptable alternative. However, in this
test, it is not as consistent as the 125-asset critical line
optimization or the optimization of the 25-asset subgroups.
The Markowitz (1959) critical line algorithm proves to be
a valuable tool for handling singular matrices. This allows
the portfolio manager to add additional constraints to the
optimization problem. Furthermore, the optimization problem can be more versatile with the addition of put option,
call option, long margin and short margin positions.
AC K N O W L E D G E M EN T
The author wishes to thank Roger Bey for providing the
Vickson and Altman (1977) correction code for the stochastic dominance program. The author is responsible for any
errors.
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